Cold-Plasma Dispersion Relation

(5.40) |

Without loss of generality, we can assume that the equilibrium magnetic field is directed along the -axis, and that the wavevector, , lies in the - plane. Let be the angle subtended between and . The eigenmode equation (5.41) can be written

The condition for a nontrivial solution is that the determinant of the square matrix be zero. With the help of the identity(5.43) |

The dispersion relation (5.44) is evidently a quadratic in , with two roots. The solution can be written

(5.48) |

(5.49) |

The dispersion relation (5.44) can also be written

For the special case of wave propagation parallel to the magnetic field (i.e., ), the previous expression reduces to Likewise, for the special case of propagation perpendicular to the field (i.e., ), Equation (5.50) yields(5.54) | ||

(5.55) |